# Two dice are thrown. The events A, B and C are as follows:

Question:

Two dice are thrown. The events A, B and C are as follows:

A: getting an even number on the first die.

B: getting an odd number on the first die.

C: getting the sum of the numbers on the dice ≤ 5

Describe the events

(i) $A^{\prime}$

(ii) not $B$

(iii) $A$ or $B$

(iv) $A$ and $B$

(v) $A$ but not $C$

(vi) $B$ or $C$

(vii) $\mathrm{B}$ and $\mathrm{C}$

(viii) $\mathrm{A} \cap \mathrm{B}^{\prime} \cap \mathrm{C}^{\prime}$

Solution:

When two dice are thrown, the sample space is given by

$\mathrm{S}=\{(x, y): x, y=1,2,3,4,5,6\}$

$=\left\{\begin{array}{llll}(1,1), & (1,2), & (1,3), & (1,4), & (1,5), & (1,6) \\ (2,1), & (2,2), & (2,3), & (2,4), & (2,5), & (2,6) \\ (3,1), & (3,2), & (3,3), & (3,4), & (3,5), & (3,6) \\ (4,1), & (4,2), & (4,3), & (4,4), & (4,5), & (4,6) \\ (5,1), & (5,2), & (5,3), & (5,4), & (5,5), & (5,6) \\ (6,1), & (6,2), & (6,3), & (6,4), & (6,5), & (6,6)\end{array}\right\}$

Accordingly,

$\mathrm{A}=\left\{\begin{array}{l}(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(4,1),(4,2),(4,3), \\ (4,4),(4,5),(4,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)\end{array}\right\}$

$B=\left\{\begin{array}{l}(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(3,1),(3,2),(3,3) \\ (3,4),(3,5),(3,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)\end{array}\right\}$

$\mathrm{C}=\{(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(3,1),(3,2),(4,1)\}$

(i) $\mathrm{A}^{\prime}=\left\{\begin{array}{l}(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(3,1),(3,2),(3,3), \\ (3,4),(3,5),(3,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)\end{array}\right\}=\mathrm{B}$

(ii) $\quad$ Not $\mathrm{B}=\mathrm{B}^{\prime}=\left\{\begin{array}{l}(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(4,1),(4,2),(4,3), \\ (4,4),(4,5),(4,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)\end{array}\right\}=\mathrm{A}$

(iii)   $A$ or $B=A \cup B=\left\{\begin{array}{llll}(1,1), & (1,2), & (1,3), & (1,4), & (1,5), & (1,6) \\ (2,1), & (2,2), & (2,3), & (2,4), & (2,5), & (2,6) \\ (3,1), & (3,2), & (3,3), & (3,4), & (3,5), & (3,6) \\ (4,1), & (4,2), & (4,3), & (4,4), & (4,5), & (4,6) \\ (5,1), & (5,2), & (5,3), & (5,4), & (5,5), & (5,6) \\ (6,1), & (6,2), & (6,3), & (6,4), & (6,5), & (6,6)\end{array}\right\}=S$

(iv) $A$ and $B=A \cap B=\phi$

(v) $\quad$ A but not $C=A-C$ $=\left\{\begin{array}{l}(2,4),(2,5),(2,6),(4,2),(4,3),(4,4),(4,5), \\ (4,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)\end{array}\right\}$

(vi) $\quad \mathrm{B}$ or $\mathrm{C}=\mathrm{B} \cup \mathrm{C}$

$=\left\{\begin{array}{l}(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2), \\ (2,3),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6), \\ (4,1),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)\end{array}\right\}$

(vii) $B$ and $C=B \cap C\{(1,1),(1,2),(1,3),(1,4),(3,1),(3,2)\}$

(viii)   $C^{\prime}=\left\{\begin{array}{l}(1,5),(1,6),(2,4),(2,5),(2,6),(3,3),(3,4),(3,5),(3,6),(4,2), \\ (4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6), \\ (6,1),(6,2),(6,3),(6,4),(6,5),(6,6)\end{array}\right\}$

$\therefore A \cap B^{\prime} \cap C^{\prime}=A \cap A \cap C^{\prime}=A \cap C^{\prime}$

$=\left\{\begin{array}{l}(2,4),(2,5),(2,6),(4,2),(4,3),(4,4),(4,5), \\ (4,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)\end{array}\right\}$